Simple model. Unlike existing models [1]}, [2]}, our empirical observation suggests that liquid-vapor phase transition is independent of the change of orientational order. Inspired by the empirical observation, we model moving individuals as small magnets with
a \(XY\) -model like interaction [3]},

U(r,)=i<jJ(rij)(i-j),


where \(r_{ij} \equiv |\textbf {r}_i-\textbf {r}_j|\) , and \(J(r)\)  is the coupling function that varies by distance. When \(J(r)\)  is the Heaviside step function, Eq. (REF ) leads to a social force similar to the Vicsek model [1]}. However, our empirical observation in Figure 1b suggests a more subtle interaction with mixed ferromagnetic and antiferromagnetic effects. Therefore, we use the Morse potential [5]} for the coupling function \(J(r)\) , which captures the empirical observation of ferromagnetic interaction at a short extent and an antiferromagnetic interaction at longer distances (Fig. 1b). In addition, we also used a soft sphere potential for each individual to take their physical size into consideration (see SM Section C).
